[Axiom-developer] [#227 'random()$Integer' is a strange function]

[unknown]

Changes http://wiki.axiom-developer.org/227RandomIntegerIsAStrangeFunction/diff
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>Since I'm currently working in statistics and probability...

In that case, I completely agree with you on exactness whenever possible 
(depending on the computation and distribution involved). But those situations 
are rare (may be limited to only some discrete random variable with a finite 
sample space like the Binomial r.v.)  I wonder how one can create even a 
normally distrihbuted random variable. In simulation programs involving a 
continuous random variable, uniformly distributed floats and the inverse of the 
cumulative distribution function are involved. How can you do any exact 
computations with continuous variables if you have to solve for the inverse of 
the cdf? You will need exact reals, and then an exact method to solve 
'cdf(x)=y', for any uniformly distributed random real number 'y' over (0,1). 
Even for discrete variables like the Poission variable, you will have to use 
floats, or else you get into exact summations and their inverses.

>But why wouldn't you use simply 'random(11)\$INT' ?

Because I was not thinking or was stupid or both!

Now that I believe I am thinking, using 'random(11)\$INT' simply passes the 
question to Lisp and if you are still picky, you should investigate how the 
Lisp function 'random(n)' is coded. Is it  some variation of the usual linear 
congruence method, but perhaps extended to arbitrary precision with a large 
base like $2^{26}$ to cover really large 'n'. For small 'n', it would be modulo 
'n' in the same way: that is, the algorithm for 'random(11)\$Lisp' may be 
actually something like  'random1(seed)$Lisp rem 11'  translated to  Lisp where 
'random1' (name is arbitrarily made up) is the Lisp random number generator for 
single precision (4 byte, for example) integers and 'seed' is the seed or 
previously generated random single precision integer.  If so, it would be only 
approximately uniform also.

>For infinite sets, we have - or should have - functions which chose a random 
>element chosen according to the distribution, indicated by the function name. 
>Of course, not all distributions make sense on all sets... But I'm certain 
>that this can be worked out.

The way to go would seem to be to create two new categories 
'ProbabilityDensityFunctions' (or 'ProbabilityDensityFunctionSpace')  and 
'RandomVariable', say 'RandomVariable(S:Set, d:ProbabilityDensityFunction)' 
where 'S' is the sample space, and 'x:=random()' would mean 'x' is a r.v. on 
'S' with probabillity density function 'd', and then create domains for these. 
The actual design, parameters, and specifications will need be more carefully 
thought out, say whether to use the cdf instead of the pdf (it seems cdf has 
lots of advantages as it is easy to recover the pdf by differentiation in many 
cases and cdf is continuous), and what are useful functions ( 'seed(x)', 
'nextRandom(x)'  come to mind, but operations on r.v. such as finding the pdf 
of joint distributions, are possible too). Then, if we were in Aldor, we can 
retroactively make 'IntegerNumberSystem' a subcategory (In Axiom, we add a 
package).  This would be a major project, since as you know, probability theory 
gets g!
eneralized into measure theory and so on. 

I think worrying about the proper syntax of 'random()' for an infinite sample 
space is the least of our (your?) problems :-)

William
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